Time-ordered product ($T\{{\mathcal{O}}_1\cdots {\mathcal{O}}_n\}$)

A time-ordered product sorts operator insertions from latest time on the left to earliest time on the right. Reordering odd fermionic operators contributes the sign of their permutation. For bosonic $A,B$, $T[A(t)B(t^{\prime })]=\theta (t-t^{\prime })A(t)B(t^{\prime })+\theta (t^{\prime }-t)B(t^{\prime })A(t)$. Vacuum expectations of free time-ordered products are evaluated by the Wick theorem.